antimatterenergyenergy-conservation
Do they create energy? Or do they just disappear with zero energy?
If they create energy when disappearing, that means it takes energy to create them, right?
If they disappear into zero energy, could that be a clue about how to create matter from nothing?
I believe this is where your thought experiment goes wrong: However, the nuclei are so small and so widely separated that presumably they just orbit each other as the electron clouds annihilate
Why would they be widely separated? Solids touch in order of nanometers. The protons and antiprotons would attract each other with the 1/r^2 electric field, which is very strong. Even if the energy of the projectile is small enough to form "protoniums", ie. have antiprotons in the place of electrons in the quantum mechanical solution, the wave function will be so large that the protons antiproton wave function will overlap and annihilate. The lifetime is of the order of .1 to 10 microseconds.
To get an idea of the annihilation Here is a bubble chamber photo of an antiproton entering from below annihilating on a proton at rest int he chamber. The multiplicity is on pions and kaons which will then decay with their characteristic decays.
The multiplicity is high even for annihilation at rest.
If the projectile has high enough energy, it will be a scattering with the cross section of antiproton on proton, depending on the area, to get the percentage annihilated. It is very difficult to get a plasma of protons and antiprotons anyway. Annihilations have a very large crossection. The antiprotons will either scatter elastically if they have enough energy, and leave the samples or annihilate if in the region of protons.
If the two bits of matter are in space with only their own gravity, then some will escape a first scatter and then fall back and annihilate or escape if they have escape velocity.
The negative energy states are called "the Dirac sea "
The Dirac sea is a theoretical model of the vacuum as an infinite sea of particles with negative energy. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, well before its experimental discovery in 1932.
Dirac sea for a massive particle. yellow• particles, blue • antiparticles
So in this model a negative energy hole appears in our positive energy world as an antiparticle , with positive mass /energy and opposite quantum numbers: the opposite to the electron is the positron.
It is a very successful model of the half spin particles we have observed.
Best Answer
The do not disappear with zero energy.
Their energy (both that originating from rest mass and any kinetic energy) appears somehow. As photons, a spray of other (lighter) particles, etc.
For instance, when an electron meets a positron (that is, a anti-electron) the most common result is a pair of gamma rays each of 511 keV (in the center of momentum (CoM) frame of the $e$--$e^+$ pair). It has to be at least two because you have to conserve both energy and momentum (and angular momentum, too, but that is a complication we will ignore for the moment), and that also tells us that there have opposite momenta in the CoM frame.